Flexible meshing type gear device with negative deflection passing tooth profile

ABSTRACT

The amount of radial flexing (w) in a basic section defined perpendicular to the axis of the flexible external gear (3) at a prescribed point on the tooth trace of the flexible external gear (3) is defined to be an amount of negative deviation flexing smaller than the normal amount of flexing (W 0 ) The rigid internal gear (2) and the flexible external gear (3) are both spur gears and the number of teeth of the flexible external gear (3) is 2n (n being a positive integer) fewer than that of the rigid internal gear (2). The working tooth profile of one of the gears (2, 3) is defined to be a convex curve whose shape is or approximates a convex curve (L2(O,B)) obtained by similarity transformation of a peak portion (L1(O,A) of the rack-approximated moving path (L1) of a tooth of the gear with respect to the other gear in the basic section of the tooth trace perpendicular to the axis, the portion being convex relative to the other gear, at an enlargement ratio (λ) using the apex of the moving path as the origin. The working tooth profile of the other gear is defined to be a concave curve whose shape is a concave curve (L3(O,C)) obtained by similarity transformation of the same portion of the moving path at an enlargement ratio (λ+1) using the apex of the moving path as the origin. The meshing of the two tooth profiles is pass messing enabling continuous contact and the ability to retain a lubricating oil film between the tooth surfaces is enhanced.

TECHNICAL FIELD

This invention relates to a flexible meshing type gear device. Moreparticularly, this invention relates to the tooth profiles of a rigidinternal gear and a flexible external gear used in a flexible meshingtype gear device.

BACKGROUND ART

A flexible meshing type gear device typically consists of a rigidcircular internal gear, a flexible external gear which has, for example,2n (n being a positive integer) fewer teeth than the internal gear andwhich is disposed inside the internal gear and flexed into an ellipticalshape so as to mesh with the internal gear at two places, and a wavegenerator fitted inside the external gear for flexing it into anelliptical shape.

Although the basic tooth profile for the gears of a flexible meshingtype gear device is linear (see U.S. Pat. No. 2,906,143), flexiblemeshing type gear devices using involute gears have also been developed(see JP-B 45-411171). In addition, the present inventor proposed asystem using as the tooth face profile of both gears the curve obtainedby similarity transforming the moving path, at a reduction ratio of 1/2,over a prescribed range from the meshing limit point on the path basedon the rack approximation of the tooth of the external gear relative tothe internal gear (JP-A 63-115943). This is a system for obtainingcontinuous contact between the tooth profiles of the tooth faces of bothgears.

One type of flexible meshing type gear device known to the art is fittedwith an annular flexible external gear and another type is fitted with acup-shaped flexible external gear. In the latter type of device, athree-dimensional flexing phenomenon called coning occurs in which theinsertion of the elliptical wave generator causes the amount of flexing(difference between the major and minor axes of the ellipse) togradually increase from the diaphragm side toward the opening portion ofthe cup-shaped flexible external gear approximately in proportion to thedistance from the diaphragm. The tooth profiles described in theaforementioned publications do not take this coning into consideration,however. Therefore, while continuous meshing of the tooth profiles ofthe two gears can be realized for tooth traces with specific sections(e.g., a non-deviated section corresponding to the normal amount offlexing), tooth interference and other problems arise at other sectionsof the tooth trace.

The inventor later proposed a flexible meshing type gear device enablinga wider mesh range, without interference, over the entire tooth trace ofthe cup-shaped flexible external gear. This device is proposed, forexample, in Japanese Patent Applications Hei 3-357036 and Hei 3-357037.

The performance being demanded of flexible meshing type gear devices isbecoming increasingly sophisticated. To respond to this demand, it isnecessary to further improve device strength and wear resistance. Thereis a particular need to improve the wear resistance of the tooth surfaceto the maximum possible.

All of the aforementioned inventions enable continuous meshing along thetooth trace. However, the meshing is so-called countermovement meshing.Since it is therefore impossible to avoid disadvantages from the pointof maintaining a lubricating oil film between the tooth surfaces, acertain limit on the permissible delivered torque is present owing totooth surface wear caused by oil film rupture. Because of this, a strongneed is felt for an improvement in this respect.

DISCLOSURE OF THE INVENTION

For achieving the aforesaid improvement, this invention replaces theprior-art tooth profile, which conducts countermovement meshinginvolving continuous contact between convex curves, with fundamentallyimproved tooth profiles for the rigid internal gear and the flexibleexternal gear. Specifically, a new convex curve tooth profile is adoptedas the working tooth profile of either the rigid internal gear or theflexible external gear and a concave curve tooth profile is adopted asthe working tooth profile of the other gear. As a result, the two gearsperform pass meshing between a convex tooth profile and a concave toothprofile, which is advantageous from the point of lubrication.

More specifically, this invention is characterized in that the followingstructural features are provided in a flexible meshing type gear devicehaving a rigid internal gear, a flexible external gear inside theinternal gear and a wave generator for flexing the section of theexternal gear perpendicular to its axis into an elliptical shape,causing the flexible external gear to mesh partially with the rigidinternal gear and rotating the mesh positions of the two gears in thecircumferential direction, the rotation of the wave generator producingrelative rotation between the two gears.

(a) The amount of radial flexing (w) in a basic section definedperpendicular to the flexible external gear axis at a prescribed pointon the tooth trace of the gear is defined to be an amount of negativedeviation flexing smaller than the normal amount of flexing (W₀)

(b) The rigid internal gear and the flexible external gear are both spurgears.

(c) The number of teeth of the flexible external gear is 2n (n being apositive integer) fewer than that of the rigid internal gear.

(d) The working tooth profile of either the rigid internal gear or theflexible external gear, designated as the first gear, is defined to be aconvex curve whose shape is or approximates a convex curve obtained bysimilarity transformation of a peak portion of the rack-approximatedmoving path of a tooth of the gear with respect to the other gear,designated as the second gear, in the basic section of the tooth traceperpendicular to the axis, the portion being convex relative to theother gear, at an enlargement ratio (λ) using the apex of the movingpath as the origin.

(e) The working tooth profile of the other or second gear is defined tobe a concave curve whose shape is or approximates a concave curveobtained by similarity transformation of the same portion of the movingpath at an enlargement ratio (λ+1) using the apex of the moving path asthe origin, whereby meshing of the two tooth profiles is pass messingenabling continuous contact in the basic sector perpendicular to theaxis.

Preferably, either the peak portion of the concave tooth profile of thesecond gear whose working tooth profile is a concave curve is formed asa convex curve in proportion to the need to avoid interference or thetooth crest is shortened.

The tooth profiles of this invention can also be applied to a flexiblemeshing type gear device equipped with a cup-shaped flexible externalgear. In order to realize continuous contact along the tooth trace inthis case, relieving is preferably applied, in proportion to the need toavoid interference, toward the opening portion of the cup-shapedflexible external gear and toward the inner end on the diaphragm sidethereof relative to the basic section of the tooth trace perpendicularto the axis.

The tooth profiles of this invention can also be applied to a flexiblemeshing type gear device in which the flexible external gear is flexedin trilobate shape to mesh with the rigid internal gear at three pointson the periphery thereof. In this case, the number of teeth of theflexible external gear is set to be 3n (n being a positive integer)fewer than the number of teeth of the rigid internal gear.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a perspective view of a flexible meshing type gear deviceequipped with a cup-shaped flexible external gear.

FIG. 2 is a schematic front view of the device of FIG. 1.

FIG. 3 is a set of diagrams for explaining how the cup-shaped flexibleexternal gear is flexed by coning, in which (a) is a section through theaxis before deformation, (b) is a section through the axis including themajor axis of the wave generator, and (c) is a section through the axisincluding the minor axis.

FIG. 4 is the moving path in a basic section perpendicular to the axis,as determined by rack approximation, in the case of negative deviationof a tooth of a flexible external gear or a rigid internal gear withrespect to the other gear.

FIG. 5 is a diagram for explaining the method of tooth profilederivation of this invention.

FIG. 6 is a diagram for explaining how the addendum portion of theconcave curve of one gear is replaced by a convex envelope.

FIG. 7 is an explanatory diagram of the meshing of the tooth profiles ofthis invention in a basic section perpendicular to the tooth, drawnrelatively with respect to one tooth space of one gear followed over thepassage of time.

FIG. 8 is an explanatory diagram of the meshing of the tooth profiles ofthis invention in a basic section perpendicular to the teeth, thediagram relating to the case where the tooth profile of the main portionof the flexible external gear is convex and the tooth profile of themain portion of the rigid internal gear is concave and being drawnspatially over all teeth of the rigid internal gear.

FIG. 9 is a spatially drawn explanatory diagram of the meshing of thetooth profiles in the case where the convexity and concavity of the mainportions of the two gears is reversed from than in FIG. 8.

FIG. 10 is a set of diagrams for explaining an example of meshinginterference of the tooth profiles of this invention in sections otherthan the basic section perpendicular to the tooth, in which (a) is for asection on the opening side of the basic section perpendicular to thetooth and (b) is for a section on the diaphragm side of the basicsection perpendicular to the tooth.

FIG. 11 is a diagram for explaining relieving of a flexible externalgear tooth.

FIG. 12 is a set of explanatory diagrams showing an example of meshingof the tooth profiles of this invention in relieved sections other thanthe basic section perpendicular to the tooth, in which (a) is for asection on the opening side of the basic section perpendicular to thetooth and (b) is for a section on the diaphragm side of the basicsection perpendicular to the tooth.

FIG. 13 is a set of explanatory diagrams showing meshing in differentunrelieved sections in the case where the tooth profile of thecup-shaped flexible external gear is corrected, in which (a) is for asection at the opening portion, (b) is for the basic section and (c) isfor a section at the inner end portion.

FIG. 14 is an explanatory diagram of the meshing of the tooth profilesof this invention in a basic section perpendicular to the teeth, thediagram relating to the case where the difference in number of teeth is3 and being drawn spatially over all teeth.

BEST MODE FOR CARRYING OUT THE INVENTION

Embodiments of the invention will be explained with reference to thedrawings in the following.

FIGS. 1 and 2 are perspective and front views of a prior-art flexiblemeshing type gear device to which this invention can be applied. Thisflexible meshing type gear device 1 comprises a cylindrical rigidinternal gear 2, a cup-shaped flexible external gear 3 disposed insidethe rigid internal gear 2, and an elliptical wave generator 4 fittedinside the cup-shaped flexible external gear 3. The cup-shaped flexibleexternal gear 3 is in a flexed state produced by the wave generator 4.The major axis and minor axis directions of the ellipse of the wavegenerator 4 are respectively designated by references symbols 4a, 4b inthe drawings.

FIG. 3 shows the flexed state in sections through the axis of thecup-shaped flexible external gear 3 caused by so-called coning, namely,by flexing the opening portion of the flexible external gear. FIG. 3(a)shows the state before deformation, (b) is a section through the axisincluding the major axis 4a of the wave generator 4, and (c) is asection through the axis including the minor axis 4b thereof. As can beseen from these diagrams, the amount of flexing produced in thecup-shaped flexible external gear 3 is maximum at the opening sidesection 3a and gradually decreases toward the inner end section 3c onthe diaphragm 3b side.

FIG. 4 is the moving path in a basic section perpendicular to the tooth(a section perpendicular to the axis used in tooth profile definitionsuch as the section taken at the center of the tooth trace indicated byIV--IV in FIG. 3(a)) of a tooth of the flexible external gear or therigid internal gear (hereinafter called the "first gear 100") withrespect to the other gear (hereinafter called the "second gear 200").The moving path L1 shown here is that in the case of so-called negativedeviation in which the amount of radial flexing (difference between thepitch circle of the flexible external gear and the maximum radiusthereof when the pitch circle is deformed into elliptical or trilobateshape) is smaller than the normal value W₀ (value obtained by dividingthe pitch circle radius of the flexible external gear by the reductionratio when the rigid internal gear is fixed), namely, in the case wherethe amount of radial flexing is κW₀, where κ is the standard flexingcoefficient (κ<1). Moreover, the moving path L1shown in the same figureis that in the case where the flexible external gear is flexed intoelliptical shape and the arrow 101 indicates the moving direction of thefirst gear 100. (Since the flexible meshing type gear device has a largenumber of teeth, the gear meshing can be approximated as that of a rackwith an infinite number of teeth. Gear meshing is therefore treated interms of rack approximation in the following discussion of tooth profilederivation with respect to this and the other figures.)

FIG. 5 is a diagram for explaining the method of tooth profilederivation of this invention. Point 0 in this figure is the apex of themoving path L1 of a tooth of the first gear (the point of maximum entryinto a tooth space of the second gear) and point V is the inflectionpoint at which the moving path L1 changes from convex to concave withrespect to the second gear. A point A is defined in the OV segment ofthe moving path L1. Taking point O as the origin (center of similarity),the curve L1(O,A) between points O and A on the moving path L1 issimilarity transformed at an enlargement ratio of λ to obtain asimulated curve L2(O,B). This curve is adopted as the working toothprofile of the first gear. Although not shown in the figure, this curveis further smoothly connected with a fillet curve. The working toothprofile of the first gear is therefore a convex tooth profile.

Again taking point O as the origin (center of similarity), the curveL1(O,A) is next similarity transformed at an enlargement ratio of (λ+1)to obtain a simulated curve L3(O,C). This curve is adopted as the basictooth profile of the second gear. It is therefore a concave toothprofile.

The value of λ is selected so that the distance of point C from point Oin the addendum direction (vertical direction in the figure) becomes thesame as or close to the amplitude OM of the path L1 in the addendumdirection. In other words, by selecting point C in this manner, λ can bedefined with respect to the earlier selected point A as

    λ≈(OC/OA)-1.

That the tooth profiles of the first and second gears with toothprofiles defined in the foregoing manner properly mesh is indicated asfollows.

In FIG. 5, take an arbitrary point P on the concave tooth profileL3(O,C) of the second gear, draw straight line OP, and define the pointsof intersection of OP with the convex tooth profile L2(O,B) of the firstgear and the moving path L1(O,A) as Q and R. In light of the process oftooth profile derivation, it holds that

    OP=(λ+1)×OR

    OQ=λ×OR.

Therefore,

    QP=OP-OQ=OR.

From the nature of the simulation, moreover, the tangents to the curvesat the three points P, Q and R are mutually parallel.

From these two facts it can be seen that the convex tooth profileL2(O,B) contacts the concave tooth profile L3(O,C) at point P when pointQ of the convex tooth profile L2(O,B) is located at point P.Specifically, continuous meshing is ensured between the convex toothprofile L2(O,B) and the concave tooth profile L3(O,C). In addition,since the meshing starts from the point where point B contacts point C(at which time point O of the first gear tooth profile is located atpoint A) and ends with meshing at point O, it is so-called pass meshing.

Actually, however, the generating action of the tooth profile of thefirst gear replaces the tooth crest portion of the second gear with aconvex envelope before the meshing reaches point A on the moving path.The situation is shown in FIG. 6. The envelope is the DE portion in thefigure. The meshing of this section is of passing type. In a flexiblemeshing type gear device equipped with a cup-shaped flexible externalgear as shown in FIG. 3, moreover, the shape of the envelope changesdepending on the location of the section perpendicular to the axis ofthe first gear and the penetration of the tooth crest into the secondgear is greatest at the inner end portion (the portion designated 3c inFIG. 3(a)).

FIG. 7 depicts the meshing of the tooth profiles of this invention in abasic section perpendicular to the tooth, relatively with respect to onetooth space of the second gear followed over the passage of time.

FIG. 8 is a spatial representation over all of the teeth, in which thefirst gear 100 is the flexible external gear 3 and the second gear 200is the rigid internal gear 2. In contrast, FIG. 9 is a representation inwhich the first gear 100 is the rigid internal gear 2 and the secondgear 200 is the flexible external gear 3.

As pointed out earlier, in a device equipped with a cup-shaped flexibleexternal gear, the shape of the curve with which the tooth crest of thefirst gear 100 envelops the second gear 200 depends on the location ofthe section perpendicular to the axis of the first gear (see FIG. 6). Inthis case, the meshing rigidity can be enhanced by defining the convextooth profile of the addendum portion of the second gear 200 as theenvelope at the inner end portion 3c. It is also possible to impart somedegree of relief to the envelope in order to accentuate wear resistance,or simply to shorten the tooth crest to obtain full pass meshing, whichis advantageous from the point of lubrication.

The nature of the tooth profile with respect to coning of the flexibleexternal gear of a flexible meshing type gear device equipped with acup-shaped flexible external gear as shown in FIG. 3 will now beconsidered. The tooth profile of this invention is derived from themoving path in a basic section and cannot be applied withoutmodification to other sections. FIG. 10 shows this taking as an examplethe case where the first gear 100 is the flexible external gear 3 andthe second gear 200 is the rigid internal gear 2. This figure is for thecase where the basic section is taken at the center of the tooth trace(at the position of the line IV--IV in FIG. 3(a)). FIG. 10(a) is for asection on the opening side of the basic section and FIG. 10(b) is for asection at the inner end on the diaphragm side of the basic section.

As can be seen from these figures, the teeth interfere in sectionstoward either side from the basic section. One method for avoiding thisis, as shown in FIG. 11, to apply opposite sides of the tooth from thebasic section with an amount of relieving matched to the amount ofinterference.

FIGS. 12(a), (b) show meshing of the tooth profiles in a section on theopening side and in a section at the inner end when relieving isapplied. Another method for achieving continuous contact along the toothtrace is to appropriately correct the tooth profile of the flexibleexternal gear without relieving.

Meshing in this case is shown in FIG. 13, wherein (a) is for a sectionat the opening portion, (b) for the basic section and (c) for a sectionat the inner end.

Although the foregoing explanation was made mainly regarding the case ofelliptical flexing, the method of the invention can be similarly appliedin the case of trilobate flexing with a difference in number of teeth of3n. FIG. 14 shows an example of the meshing in the basic sector in thiscase, spatially over all of the teeth.

Industrial Applicability

As set out in the foregoing, by introducing pass meshing the presentinvention makes it possible to greatly enhance the ability to retain alubricating oil film between the tooth surfaces and to markedly improvethe permissible transmitted torque of the flexible meshing type geardevice based on tooth surface wear.

Moreover, the adoption of negative deviation reduces the bending stressproduced by flexing in the vicinity of the major and minor axes of theflexible external gear, enhances the rim strength of the flexibleexternal gear, and enables an improvement in meshing rigidity owing tocontinuous tooth contact.

In addition, since the invention can be implemented at an arbitraryconing angle of the cup-shaped flexible external gear, it can also beapplied to a flexible external gear with a short body. It is furtherapplicable to an annular flexible external gear with no coning.

I claim:
 1. A flexible meshing type gear device with negative deflection passing tooth profile having a rigid internal gear, a flexible external gear inside the internal gear and a wave generator for flexing a section of the external gear perpendicular to its axis into an elliptical shape, causing the flexible external gear to mesh partially with the rigid internal gear and rotating mesh positions of the two gears in a circumferential direction, the rotation of the wave generator producing relative rotation between the two gears, characterized in being provided with the following structural features:(a) an amount of radial flexing (w) in a basic section defined perpendicular to a flexible external gear axis at a prescribed point on a tooth trace of the gear is defined to be an amount of negative deviation flexing smaller than a normal amount of flexing (W₀), (b) the rigid internal gear and the flexible external gear are both spur gears, (c) a number of teeth of the flexible external gear is 2n (n being a positive integer) fewer than that of the rigid internal gear, (d) a working tooth profile of either one of the rigid internal gear or the flexible external gear, called a first gear, is defined to be a convex curve whose shape approximates a convex curve obtained by similarity transformation of a peak portion of a rack-approximated moving path of a tooth of the gear with respect to the other gear, called a second gear, in the basic section of the tooth trace perpendicular to the axis, the portion being convex relative to the second gear, at an enlargement ratio (λ) using an apex of the moving path as an origin, and (e) the working tooth profile of the second gear is defined to be a concave curve whose shape approximates a concave curve obtained by similarity transformation of the same portion of the moving path at an enlargement ratio (λ+1) using the apex of the moving path as the origin, whereby meshing of the tooth profiles of the first and second gears is pass messing enabling continuous contact in the basic sector perpendicular to the axis.
 2. A flexible meshing type gear device with negative deflection passing tooth profile according to claim 1, wherein the peak portion of the concave tooth profile of the second gear whose working tooth profile is a concave curve is defined by a convex curve to avoid interference of the peak portion of the concave tooth profile with the convex first gear tooth profile defined from a convex curve.
 3. A flexible meshing type gear device with negative deflection passing tooth profile according to claim 1, wherein a tooth crest of the second gear, whose working tooth profile is a concave curve, is shortened to avoid interference of the tooth crest of the concave tooth profile with the convex first gear tooth profile defined from a convex curve.
 4. A flexible meshing type gear device with negative deflection passing tooth profile having a rigid internal gear, a flexible external gear inside the internal gear and a wave generator for flexing a section of the external gear perpendicular to its axis into a trilobate shape, causing the flexible external gear to mesh partially with the rigid internal gear and rotating mesh positions of the two gears in a circumferential direction, the rotation of the wave generator producing relative rotation between the two gears, characterized in being provided with the following structural features:(a) an amount of radial flexing (w) in a basic section defined perpendicular to a flexible external gear axis at a prescribed point on a tooth trace of the gear is defined to be an amount of negative deviation flexing smaller than a normal amount of flexing (W₀), (b) the rigid internal gear and the flexible external gear are both spur gears, (c) a number of teeth of the flexible external gear is 3n (n being a positive integer) fewer than that of the rigid internal gear, (d) a working tooth profile of either one of the rigid internal gear or the flexible external gear, called a first gear, is defined to be a convex curve whose shape approximates a convex curve obtained by similarity transformation of a peak portion of a rack-approximated moving path of a tooth of the gear with respect to the other, called a second gear, in the basic section of the tooth trace perpendicular to the axis, the portion being convex relative to the second gear, at an enlargement ratio (λ) using an apex of the moving path as an origin, and (e) the working tooth profile of the second gear is defined to be a concave curve whose shape approximates a concave curve obtained by similarity transformation of the same portion of the moving path at an enlargement ratio (λ+1) using the apex of the moving path as the origin, whereby meshing of the tooth profiles of the first and second gears is pass messing enabling continuous contact in the basic sector perpendicular to the axis.
 5. A flexible meshing type gear device with negative deflection passing tooth profile according to claim 4, wherein the peak portion of the concave tooth profile of the second gear whose working tooth profile is a concave curve is defined by a convex curve to avoid interference of the peak portion of the concave tooth profile with the convex first gear tooth profile defined from a convex curve.
 6. A flexible meshing type gear device with negative deflection passing tooth profile according to claim 4, wherein a tooth crest of the second gear, whose working tooth profile is a concave curve, is shortened to avoid interference of the tooth crest of the concave tooth profile with the convex first gear tooth profile defined from a convex curve.
 7. A flexible meshing type gear device with negative deflection passing tooth profile having a rigid internal gear, a cup-shaped flexible external gear inside the internal gear and a wave generator for flexing the external gear into an elliptical shape in a section perpendicular to its axis such that an amount of flexing is produced from a diaphragm end to an opening end thereof in approximate proportion to distance from the diaphragm, causing the external gear to mesh partially with the rigid internal gear and rotating mesh positions of the two gears in a circumferential direction, the rotation of the wave generator producing relative rotation between the two gears, characterized in being provided with the following structural features:(a) an amount of radial flexing (w) in a basic section defined perpendicular to a flexible external gear axis at a prescribed point on a tooth trace of the gear is defined to be an amount of negative deviation flexing smaller than a normal amount of flexing (W₀), (b) the rigid internal gear and the flexible external gear are both spur gears, (c) a number of teeth of the flexible external gear is 2n (n being a positive integer) fewer than that of the rigid internal gear, (d) a working tooth profile of either one of the rigid internal gear or the flexible external gear, called a first gear, is defined to be a convex curve whose shape is or approximates a convex curve obtained by similarity transformation of a peak portion of a rack-approximated moving path of a tooth of the gear with respect to the other gear, called a second gear, in the basic section of the tooth trace perpendicular to the axis, the portion being convex relative to the second gear, at an enlargement ratio (λ) using an apex of the moving path as an origin, and (e) the working tooth profile of the second gear is defined to be a concave curve whose shape approximates a concave curve obtained by similarity transformation of the same portion of the moving path at an enlargement ratio (λ+1) using the apex of the moving path as the origin, whereby meshing of the tooth profiles of the first and second gears is pass messing enabling continuous contact in the basic sector perpendicular to the axis.
 8. A flexible meshing type gear device with negative deflection passing tooth profile according to claim 7, wherein the peak portion of the concave tooth profile of the gear whose working tooth profile is a concave curve is defined by a convex curve to avoid interference of the peak portion of the concave tooth profile with the convex tooth profile defined from a convex curve.
 9. A flexible meshing type gear device with negative deflection passing tooth profile according to claim 7, wherein a tooth crest of the second gear, whose working tooth profile is a concave curve, is shortened to avoid interference of the tooth crest of the concave tooth profile with the convex first gear tooth profile defined from a convex curve.
 10. A flexible meshing type gear device with negative deflection passing tooth profile according to claim 8, wherein the gear profile of at least one of the first and second gears is applied with relieving toward the opening portion of the cup-shaped flexible external gear and toward the inner end on a diaphragm side thereof relative to the basic section of the tooth trace perpendicular to the axis, to avoid interference between the two tooth profiles owing to coning of the cup-shaped flexible external gear.
 11. A flexible meshing type gear device with negative deflection passing tooth profile having a rigid internal gear, a cup-shaped flexible external gear inside the internal gear and a wave generator for flexing the external gear into a trilobate shape in a section perpendicular to its axis such that an amount of flexing is produced from a diaphragm end to an opening end thereof in approximate proportion to distance from the diaphragm, causing the external gear to mesh partially with the rigid internal gear and rotating mesh positions of the two gears in a circumferential direction, the rotation of the wave generator producing relative rotation between the two gears, characterized in being provided with the following structural features:(a) an amount of radial flexing (w) in a basic section defined perpendicular to a flexible external gear axis at a prescribed point on a tooth trace of the gear is defined to be an amount of negative deviation flexing smaller than a normal amount of flexing (W₀), (b) the rigid internal gear and the flexible external gear are both spur gears, (c) a number of teeth of the flexible external gear is 3n (n being a positive integer) fewer than that of the rigid internal gear, (d) a working tooth profile of either one of the rigid internal gear or the flexible external gear, called a first gear, is defined to be a convex curve whose shape approximates a convex curve obtained by similarity transformation of a peak portion of a rack-approximated moving path of a tooth of the gear with respect to the other gear, called a second gear, in the basic section of the tooth trace perpendicular to the axis, the portion being convex relative to the second gear, at an enlargement ratio (λ) using an apex of the moving path as an origin, and (e) the working tooth profile of the second gear is defined to be a concave curve whose shape approximates a concave curve obtained by similarity transformation of the same portion of the moving path at an enlargement ratio (λ+1) using the apex of the moving path as the origin, whereby meshing of the tooth profiles of the first and second gears is pass messing enabling continuous contact in the basic sector perpendicular to the axis.
 12. A flexible meshing type gear device with negative deflection passing tooth profile according to claim 11, wherein the peak portion of the concave tooth profile of the second gear whose working tooth profile is a concave curve is defined by a convex curve to avoid interference of the peak portion of the concave tooth profile with the convex first gear tooth profile defined from a convex curve.
 13. A flexible meshing type gear device with negative deflection passing tooth profile according to claim 11, wherein a tooth crest of the first gear, whose working tooth profile is a concave curve, is shortened to avoid interference of the tooth crest of the concave tooth profile with the convex first gear tooth profile defined from a convex curve.
 14. A flexible meshing type gear device with negative deflection passing tooth profile according to claim 12, wherein the gear profile of at least one of the first and second gears is applied with relieving toward the opening portion of the cup-shaped flexible external gear and toward the inner end on a diaphragm side thereof relative to the basic section of the tooth trace perpendicular to the axis, to avoid interference between the two tooth profiles owing to coning of the cup-shaped flexible external gear. 